Symplectic tracing of high‐energy charged particles in the inner magnetosphere

Abstract

To investigate the radiation belt, the Gauss Runge‐Kutta method (RKG), one of the symplectic numerical integration methods, is introduced and tested by tracing protons and electrons in geomagnetic dipole field. The tracings of protons having the energies from 10 keV to 10 MeV with pitch angles from 90° to 30° at L ∼ 5 are practiced by fourth‐ and sixth‐order RKG with time step 0.016 of the Larmor period on the equatorial plane. The accuracy of calculations are examined with Cartesian, cylindrical, and polar coordinates. With all coordinates, we confirmed that the calculations are stable for 10,000,000 steps with RKG. On the other hand, the calculations for protons in the same conditions with standard Runge‐Kutta methods fail within 10,000 steps. The tracings of relativistic electrons having energies from 10 to 150 MeV with pitch angles from 90° to 30° at L ∼ 5 are also practiced. For the electrons above 50 MeV, the calculations are successful for 10,000,000 steps with the time step 0.1 × RE/c by fourth and sixth RKG with cylindrical coordinates. The calculations by standard Runge‐Kutta method are failed in any coordinate system, and the calculations by fourth or sixth RKG with Cartesian or polar coordinates are also failed for all energy range above. The calculations for the electrons below 10 MeV with this time step size are all failed by any methods and with any coordinate systems mentioned above. For the effective calculations, for example, for 10 MeV electron with pitch angle 90° by fourth order RKG, we must use shorter timescale size than 0.2 of the above size.